For an -digit decimal integer , moving its leading digit to the end gives a cyclic rotation satisfyingThus, when a modulus divides , every rotation has the same divisibility by as , up to multiplication by a power of ten.
Equality holds exactly when the multiplicative order divides , so one period contains no repeated values.
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The multiplicative order of an integer \( a \) modulo \( n \) is defined as the smallest positive integer \( k \) such that \[ a^k \equiv 1 \mod n. \] In simpler terms, it is the smallest exponent \( k \) for which raising \( a \) to the power of \( k \) results in a value that, when divided by \( n \), leaves a remainder of 1.